条件最优输运中相干成本的曲率界

中文摘要 AI 辅助

我们研究了给定两参数族条件概率测度的联合粒子实现中，集成方向最小动能与最小期望Dirichlet能量之间的差距。在具有紧连通纤维和星形参数域的正则余面积假设下，我们推导了一个曲率下界，该下界对实现给定光滑正条件测度的所有可容许Sobolev映射律均成立。我们的方法将局部最优输运速度的加权Poisson刻画与精确能量残差恒等式、弱Stokes公式及显式输运构造相结合。当且仅当Lie括号曲率在整个参数域上消失时，最小超额恰好为零；而在非零曲率中心，对于半边长$r$且具有非归一化面积测度的正方形，当$r\ o0$时该超额为$r^4$阶。在拟合电力需求数据的两相环面模型中，数值测试展示了显式耦合的该四次阶，径向输运在测试的小正方形上大约将有序构造的超额减半。

英文摘要

We study the gap between the integrated direction-wise kinetic minima and the minimum expected Dirichlet energy of a joint particle realization of a prescribed two-parameter family of conditional probability measures. Under regular coarea assumptions with compact connected fibres and a star-shaped parameter domain, we derive a curvature lower bound valid for every admissible law of Sobolev maps realizing the prescribed smooth positive conditional measures. Our method combines the weighted-Poisson characterization of local optimal-transport velocities with an exact energy-residual identity, a weak Stokes formula, and explicit transport constructions. The minimum excess vanishes exactly when the Lie-bracket curvature vanishes throughout the parameter domain, whereas at a nonzero-curvature centre it is of order $r^4$ as $r\to0$ on squares of half-side length $r$ with unnormalized area measure. In a two-phase torus model fitted to electricity-demand data, numerical tests illustrate this quartic order for explicit couplings, with radial transport approximately halving the ordered construction's excess on the tested small squares.

发表机构

• Centre de Mathématiques Appliquées (CMAP), École Polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院，应用数学中心）
• Sorbonne Université（索邦大学）

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